Problem: One endpoint of a line segment is $(4,3)$ and its midpoint is $(2,9)$. What is the sum of the coordinates of the other endpoint?
Let the other endpoint be $(x, y)$. We know that $\frac{4 + x}{2} + \frac{3 + y}{2} = 2 + 9 = 11$. Thus, $7 + x + y = 22$, and $x + y = \boxed{15}$.